On Subsystem Codes Beating the Hamming or Singleton Bound

Abstract

Subsystem codes are a generalization of noiseless subsystems, decoherence free subspaces, and quantum error-correcting codes. We prove a Singleton bound for GF(q)-linear subsystem codes. It follows that no subsystem code over a prime field can beat the Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the Hamming bound. A number of open problems concern the comparison in performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal MDS stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.

18 pages more densely packed than classically possible

References in corpus (6)

• Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
• Stabilizer Formalism for Operator Quantum Error Correction
• A Unified and Generalized Approach to Quantum Error Correction
• Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes
• On Protected Realizations of Quantum Information
• Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes

Cited by in corpus (8)

• Entropic proofs of Singleton bounds for quantum error-correcting codes
• Generalized Multiplicative Domains and Quantum Error Correction
• Singleton Bounds for Entanglement-Assisted Classical and Quantum Error Correcting Codes
• Encoding Classical Information in Gauge Subsystems of Quantum Codes
• Reducing circuit depth with qubitwise diagonalization
• Constructions of Subsystem Codes over Finite Fields
• Asymptotics of the quantum Hamming bound for subsystem codes
• Bounds on concatenated entanglement-assisted quantum error-correcting codes